Game Theoretic Pragmatics
Anton Benz
Centre for General Linguistics, Berlin
ESSLLI 2010
Copenhagen, 16 August – 20 August 2010
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 1 / 70
The Course
1. addresses topics of Gricean Pragmatics.2. concentrates mainly on two frameworks:
i. Iterated Best Responseii. Optimal Answer Model
3. is based on classical game theory.
4. not concerned with the evolution of language structure and use.⇒ no evolutionary game theory!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 2 / 70
Models of Signalling Behaviour
GT Models
non–evolutionary evolutionary
replicationinvolvingseveralgenerations
largepopulations(infinite)
online competence
immediateeffects
twointerlocutors
several roundof interaction
twointerlocutors
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 3 / 70
Models of Signalling Behaviour
GT Models
non–evolutionary evolutionary
replicationinvolvingseveralgenerations
largepopulations(infinite)
online competence
immediateeffects
twointerlocutors
several roundof interaction
twointerlocutors
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 3 / 70
Outline
1. Introduction and Motivation2. The Basic Iterated Best Response Model3. The Basic Optimal Answer Model4. Aspects of Bounded Rationality5. Some Extensions of the Optimal Answer Model
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 4 / 70
Game Theoretic PragmaticsDay 1
Introduction and Motivation
Anton Benz
Centre for General Linguistics (ZAS), Berlin
16 August 2010
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 5 / 70
Outline
1 Gricean Pragmatics and Game Theory
2 Game and Decision Theory
3 Why a New Framework
4 A Graphical Solution to the Out–of–Petrol Example
5 Introducing Signalling Games
6 Parikh’s Example: Resolving Ambiguities
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 6 / 70
Gricean Pragmatics and Game Theory
Section 1
Gricean Pragmatics and GameTheory
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 7 / 70
Gricean Pragmatics and Game Theory
A simple picture of communication
1. The speaker encodes some proposition p.2. He sends it to an addressee.3. The addressee decodes it again and writes p in his
knowledgebase.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 8 / 70
Gricean Pragmatics and Game Theory
Problem
⇒ Problem: We often communicate much more than we literally say!
Some students failed the exam.+> Most of the students passed the exam.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 9 / 70
Gricean Pragmatics and Game Theory
Communicated MeaningHerbert Paul Grice (1913–1988), see [Grice, 1989]
Grice distinguishes between:
What is said.What is implicated.
Example 1“Some of the boys came to the party.”
said: at least two cameimplicated: not all came
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 10 / 70
Gricean Pragmatics and Game Theory
Assumptions about Conversation
Conversation is a cooperative effort;Each participant recognises a common purpose in the talkexchange.
Example 2A stands in front of his obviously immobilised car.A: I am out of petrol.B: There is a garage round the corner.
⇒ Joint purpose of B’s response: Solve A’s problem of finding petrolfor his car.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 11 / 70
Gricean Pragmatics and Game Theory
Implicatures[Grice, 1989, p. 86]
What is an implicature?“. . . what is implicated is what is required that one assume aspeaker to think in order to preserve the assumption that he isobserving the Cooperative Principle (and perhaps someconversational maxims as well), . . . ”
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 12 / 70
Gricean Pragmatics and Game Theory
The Cooperative Principle
The Cooperative Principle“Make your conversational contribution such as is required, at thestage at which it occurs, by the accepted purpose or direction of thetalk exchange in which you are engaged.” [Grice, 1989]
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 13 / 70
Gricean Pragmatics and Game Theory
The Conversational Maxims
1. The Maxim of Quality: Try to make your contribution one that istrue, specifically:
1 Do not say that you believe to be false.2 Do not say that for which you lack adequate evidence.
2. The Maxim of Quantity:1 Make your contribution as informative as is required by the current
purpose of the exchange.2 Do not make your contribution more informative than is required.
3. The Maxim of Relation: Make your contribution relevant.4. The Maxim of Manner: Be perspicuous, and specifically:
1 avoid obscurity,2 avoid ambiguity,3 be brief,4 be orderly.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 14 / 70
Gricean Pragmatics and Game Theory
Examples of Implicatures
(Quantity:)1. John has five children.
+> John has not more than five children.2. The flag is white.
+> The flag is white all over.(Relevance:)
3. A: Smith doesn’t seem to have a girlfriend these days.B: He has been paying a lot of visits to New York lately.+> Smith presumably has a girlfriend in New York.
4. A is writing a testimonial about a pupil who is a candidate for aphilosophy job. A writes: “Dear Sir, Mr. X’s command of English isexcellent, and his attendance at tutorials has been regular. Yoursetc.”+> Mr. X is no good in philosophy.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 15 / 70
Gricean Pragmatics and Game Theory
Examples of Implicatures
(Manner:)5. Open the door!
Walk up to the door, turn the door handle clockwise as far as it willgo, and then pull gently toward you.+> Pay special attention to what you are doing!
6. Miss Singer sang an aria from Rigoletto.Miss Singer produced a series of sounds corresponding closely tothe score of an aria from Rigoletto.+> Miss Singer’s performance was very bad.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 16 / 70
Gricean Pragmatics and Game Theory
The Conversational Maximsshort, without Manner
Maxim of Quality: Be truthful!Maxim of Quantity:
Say as much as you can.Say no more than you must.
Maxim of Relevance: Be relevant!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 17 / 70
Gricean Pragmatics and Game Theory
The Conversational Maximsvery short, without Manner
(QQR)Be truthful (Quality) and say as much as you can(Quantity) as long as it is relevant (Relevance).
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 18 / 70
Gricean Pragmatics and Game Theory
An ApplicationA Case of a Scalar Implicature
Example 3“Some of the boys came to the party.”
said: at least two cameimplicated: not all came
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 19 / 70
Gricean Pragmatics and Game Theory
An Explanation based on Maxims
Let A(x) ≡ “x of the boys came to the party.”1. The speaker had the choice between the forms A(all) and
A(some).2. A(all) is more informative than A(some) and the additional
information is also relevant.3. Hence, if all of the boys came, then A(all) is preferred over
A(some) (Quantity) + (Relevance).
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 20 / 70
Gricean Pragmatics and Game Theory
4. The speaker said A(some).5. Hence it cannot be the case that all came.6. Therefore some but not all came to the party.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 21 / 70
Gricean Pragmatics and Game Theory
A Graphical InterpretationSituation: All of the boys came to the party.
1. The speaker has a choice between A(all) and A(some).2. If he chooses A(all), the hearer has to interpret all by the universal
quantifier.3. If he chooses A(some), the hearer has to interpret some by the
existential quantifier.
u1
PPPPPPPPq
u
u
-
-
A(all)
A(some)
∀
∃
u
uAnton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 22 / 70
Gricean Pragmatics and Game Theory
Adding Alternative SituationAlternative Situation: Some but not all came.
4. If he chooses A(some), the hearer has to interpret some by theexistential quantifier.
u1
PPPPPPPPq
u
u
-
-
A(all)
A(some)
∀
∃∀
u
uu - u - uA(some) ∃
∃¬∀
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 23 / 70
Gricean Pragmatics and Game Theory
Adding Speaker’s Preferences
u1
PPPPPPPPq
u
u
-
-
A(all)
A(some)
∀
∃
∀
u
uu - u - uA(some) ∃
∃¬∀
1
0
1
6
(Quantity)
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 24 / 70
Gricean Pragmatics and Game Theory
Adding Speaker’s Preferences
u1
PPPPPPPPq
u
u
-
-
A(all)
A(some)
∀
∃
∀
u
uu - u - uA(some) ∃
∃¬∀
1
0
1
6
(Quantity)
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 24 / 70
Gricean Pragmatics and Game Theory
Simplifying the TreeEliminate all dominated speaker’s choices
After elimination of all branches which the speaker will not choose:
u - u -A(all) ∀
∀ uu - u - uA(some) ∃
∃¬∀
1
1
HHHHe is in this situation!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 25 / 70
Gricean Pragmatics and Game Theory
Simplifying the TreeEliminate all dominated speaker’s choices
Hence, the hearer can infer from an utterance of A(some):
u - u -A(all) ∀
∀ uu - u - uA(some) ∃
∃¬∀
1
1
HHHHe is in this situation!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 25 / 70
Game and Decision Theory
Section 2
Game and Decision Theory
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 26 / 70
Game and Decision Theory
Remark
The situation depicted in the graph for scalar implicatures the outcomedepends on the decision of the speaker only!
Decision theory: decisions of individual agentsGame theory: interdependent decisions of several agents
⇒ Choice of optimal speaker strategy was a problem of decisiontheory!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 27 / 70
Game and Decision Theory
Decision Theory
If a decision only depends on
the state of the world,the actions to choose from andtheir outcomes
but not onthe choice of actions by other agents,
then the problem belongs to decision theory.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 28 / 70
Game and Decision Theory
Game TheoryA game is being played by a group of individuals whenever the fate ofan individual in the group depends not only on his own actions but alsoon the actions of the rest of the group. [Binmore, 1990]
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 29 / 70
Game and Decision Theory
Game Theory and Pragmatics
In a very general sense we can say that we play a game togetherwith other people whenever we have to decide between severalactions such that the decision depends on:
the choice of actions by othersour preferences over the ultimate results.
Whether or not an utterance is successful depends onhow it is taken up by its addresseethe overall purpose of the current conversation.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 30 / 70
Why a New Framework
Section 3
Why a New Framework
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 31 / 70
Why a New Framework
Why a New Framework?
Basic concepts of Gricean pragmatics areundefined, most notably the concept of relevance.On a purely intuitive level, it is often not possible todecide whether an inference of an implicature iscorrect or not.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 32 / 70
Why a New Framework
Out-of-Petrol Example
A stands in front of his obviously immobilised car:
A: I am out of petrol.B: There is a garage round the corner. (G)+> The garage is open. (R)
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 33 / 70
Why a New Framework
A possible Explanation
Set R∗ := The negation of R.
1. B said that G but not that R∗.2. R∗ is relevant and G ∧ R∗ ⇒ G.3. Hence, if G ∧ R∗, then B should have said G ∧ R∗
(Quantity).4. Hence, R∗ cannot be true, and therefore R.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 34 / 70
Why a New Framework
Problem: A Second Valid ExplanationExchange R and R∗
1. B said that G but not that R.2. R is relevant and G ∧ R ⇒ G.3. Hence, if G ∧ R, then B should have said G ∧ R
(Quantity).4. Hence, R cannot be true, and therefore R∗.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 35 / 70
Why a New Framework
Problem
As in the second step both R and R∗ can becalled relevant, it cannot be decided whichexplanation is correct.All decision theoretic standard measures ofrelevance would predict that R is relevant, andhence that R∗ should be implicated (which iswrong).
⇒ Without definition of relevance it is not possible todecides whether a typical explanation of arelevance implicature is in fact valid or not.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 36 / 70
A Graphical Solution to the Out–of–Petrol Example
Section 4
A Graphical Solution to theOut–of–Petrol Example
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 37 / 70
A Graphical Solution to the Out–of–Petrol Example
Out-of-Petrol ExampleModified version
Example 4A stands in front of his obviously immobilised car:
A: I am out of petrol.B: There is a garage to the left round the corner. (Gl )
Possible alternative:There is a garage to the right round the corner. (Gr )
Properties:Garage can be open and closed.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 38 / 70
A Graphical Solution to the Out–of–Petrol Example
First Steps Towards a ModelWorld Garage Garage Action Action random search
left right gl gr rw1 open open 1 1 εw2 open closed 1 0 εw3 open — 1 0 εw4 closed open 0 1 εw5 closed closed 0 0 εw6 closed — 0 0 εw7 — open 0 1 εw8 — closed 0 0 εw9 — — 0 0 ε
Meaning:open: at this place there is a garage and it is open.closed: at this place there is a garage and it is closed.— : at this place there is no garage.Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 39 / 70
A Graphical Solution to the Out–of–Petrol Example
Some Simplifications for the Graphical Solution
Next, we provide a graphical solution.We simplify the trees by considering the following worlds only:
World Garage Garage Probab. Action Action randomleft right gl gr search
rv = w2 open closed ρ 1 0 εw = w4 closed open ρ′ 0 1 ε
In the first tree, we simplify further by omitting random search r .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 40 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game Tree
ρ
ρ′
v
w
Gl
Gr
1
Gl
Gr
1
glgr
glgr
glgr
glgr
glgr
glgr
101010
010101
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 41 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreeHearer chooses optimal act.
ρ
ρ′
v
w
Gl
Gr
1
Gl
Gr
1
gl
gr
r
gl
gr
r
1
0
ε
0
1
ε
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 42 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreeSpeaker calculating backwards
ρ
ρ′
v
w
Gl
Gr
1
Gl
Gr
1
1
0
ε
0
1
ε
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 43 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreeSpeaker choosing optimal action
ρ
ρ′
v
w
Gl
Gr
1
1
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 44 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreePredicted behaviour
ρ
ρ′
v
w
Gl
Gr
H knows he is hereimplicates
gl
gr
1
1
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 45 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreeHearer can infer that the speaker is in v when uttering Gl !
ρ
ρ′
v
w
Gl
Gr
H knows he is here
implicates
gl
gr
1
1
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 45 / 70
A Graphical Solution to the Out–of–Petrol Example
The Game TreeHearer can infer that the speaker is in v when uttering Gl !
ρ
ρ′
v
w
Gl
Gr
H knows he is here
implicates
gl
gr
1
1
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 45 / 70
A Graphical Solution to the Out–of–Petrol Example
Assumptions
Actions, worlds, probabilities, and utilities:
World Garage Garage Probability Action Action randomleft right gl gr search
rv open closed ρ 1 0 εw closed open ρ′ 0 1 ε
Implicit Assumptions:Before learning anything: H chooses random search r .After learning that there is garage to the left: H chooses gl .After learning that there is garage to the right: H chooses gr .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 46 / 70
A Graphical Solution to the Out–of–Petrol Example
Collecting the Elements
In our graphical model, we represented:WorldsActionsUtilitiesProbabilities
In addition, we like to have representations of:Information states of interlocutors.Decision rules for choosing between actions.The speaker’s and hearer’s strategies.
⇒ Introduce Signalling Games!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 47 / 70
Introducing Signalling Games
Section 5
Introducing Signalling Games
[Benz et al., 2006]
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 48 / 70
Introducing Signalling Games
Static v.s. Dynamic GamesSome basic distinctions in game theory
Static game: In a static game every player performs only oneaction, and all actions are performed simultaneously.Dynamic game: In dynamic game there is at least one possibilityto perform several actions in sequence.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 49 / 70
Introducing Signalling Games
Normal Form v.s. Extensive FormSome basic distinctions in game theory
Normal form: Representation in matrix form.Extensive form: Representation in tree form. It is more suitablefor dynamic games.
The most important games for us are signalling games.They will be represented in extensive form.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 50 / 70
Introducing Signalling Games
Playing a Signalling Game
A signalling game is played in the following order:1. Nature chooses a world with a certain probability.2. An information state (his type) is assigned to each interlocutor.3. The game starts with a message sent by the speaker.4. After receiving the message, the hearer chooses an action from
his action set.5. This ends the game.
An interlocutor’s type represents his private knowledge.All other parameters of the game are assumed to be commonknowledge.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 51 / 70
Introducing Signalling Games
The Game Tree
1. The game tree shows three sequential moves: Nature, Speaker,Hearer, and their final payoffs.
2. The speaker’s type θS and the hearer’s type θH are assigned to thenodes at which they have to act.
3. The edges are labelled by the moves or acts of the players.
S HNature Outcome
θS
θS
θH
θ′H
θH
1,10,11,00,00,11,0. . .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 52 / 70
Introducing Signalling Games
The Game Tree
1. The game tree shows three sequential moves: Nature, Speaker,Hearer, and their final payoffs.
2. The speaker’s type θS and the hearer’s type θH are assigned to thenodes at which they have to act.
3. The edges are labelled by the moves or acts of the players.
S HNature Outcome
θS
θS
θH
θ′H
θH
1,10,11,00,00,11,0. . .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 52 / 70
Introducing Signalling Games
The Game Tree
1. The game tree shows three sequential moves: Nature, Speaker,Hearer, and their final payoffs.
2. The speaker’s type θS and the hearer’s type θH are assigned to thenodes at which they have to act.
3. The edges are labelled by the moves or acts of the players.
S HNature Outcome
θS
θS
θH
θ′H
θH
v
w
FF ′
F
ab
ab
ab
1,10,11,00,00,11,0. . .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 52 / 70
Introducing Signalling Games
The Game Tree
1. Two nodes are indiscernible to the speaker if they are assignedthe same type θS.
2. Two nodes are indiscernible to the hearer if they are assigned thesame type θH and the same signal.
3. Empty.
S HNature Outcome
θS
θS
θH
θ′H
θH
v
w
FF ′
F
ab
ab
ab
1,10,11,00,00,11,0. . .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 53 / 70
Introducing Signalling Games
The Game Tree
1. Two nodes are indiscernible to the speaker if they are assignedthe same type θS.
2. Two nodes are indiscernible to the hearer if they are assigned thesame type θH and the same signal.
3. Empty.
S HNature Outcome
θS
θS
θH
θ′H
θH
v
w
FF ′
F
ab
ab
ab
1,10,11,00,00,11,0. . .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 53 / 70
Introducing Signalling Games
Probablilities
1. P(v , θS, θH): Probability with which nature chooses world v ,speaker type θS, and hearer type θH .
2. We can think of P(v , θS, θH) as the result of first choosing v , andthen simultaneously θS and θH :
P(v , θS, θH) = P(v)× P(θS, θH |v) (5.1)
⇒ Collecting all these elements in a structure leads to signallinggames.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 54 / 70
Introducing Signalling Games
Signalling GamesGeneral Form
Definition 5 (Signalling game)
A signalling game is a tuple 〈Ω,ΘS,ΘH ,P,F ,A,uS,uH〉 with:1. Ω: A set of possible worlds.2. ΘS,ΘH : two finite set of types for the speaker S and the hearer H.3. P: a probability measure on Ω×ΘS ×ΘH ;4. F : a set of signals from which the speaker S chooses his
utterance.5. A: the set of actions from which the hearer H chooses his action.6. uS,uH : payoff functions which map sequences〈v ,F ,a〉 ∈ Ω×F ×A to real numbers.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 55 / 70
Introducing Signalling Games
Payoffs
1. The payoff functions uS,uH represent the preferences of theinterlocutors over outcomes of their interaction.
2. We will mostly assume that the payoff functions are provided by ajoint payoff function u.
3. That the payoff function is joint means that the preferences ofspeaker and hearer are identical.
⇒ It is a games of pure coordination!
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 56 / 70
Introducing Signalling Games
Types
1. Types may be arbitrary objects. They don’t have intrinsic meaning.2. The information set of an agent is defined by an indiscernibility
relation between tree nodes.3. That we chose Ω for the hearer was for purely mnemotechnical
reasons. We could have chosen any other object as well.4. If the hearer’s type is the same for all possibilities, then the hearer
has no private knowledge.5. In this case, we can simplify the game by eliminating the hearer’s
types.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 57 / 70
Introducing Signalling Games
Pure Strategies in a Signalling Game
1. Pure strategies are functions from information sets into actionsets.
2. The speaker’s information set is defined by his type θS.3. The hearer’s information set is defined by his type θH and the
speaker’s previous message F ∈ F .
Pure Strategies:
S : ΘS −→ F and H : ΘH ×F −→ A.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 58 / 70
Introducing Signalling Games
Mixed Strategies in a Signalling Game
1. Mixed strategies are functions from information sets into the set ofprobability distributions over an action set.
2. The information sets do not change.
We write:S(F |θS): the probability with which the speaker sends the form Fgiven type θS.H(a|θH ,F ): the probability with which the hearer chooses action agiven type θH and message F .
If there is only one hearer type, the hearer’s mixed strategy is ofthe form H( . |F ).
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 59 / 70
Parikh’s Example: Resolving Ambiguities
Section 6
Parikh’s Example: ResolvingAmbiguities
[Parikh, 2001]
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 60 / 70
Parikh’s Example: Resolving Ambiguities
Resolving Ambiguities
Example 6 (Parikh’s standard example)1. Every ten minutes a man gets mugged in New York. (A)2. Every ten minutes some man or other gets mugged in New York.
(F )3. Every ten minutes a particular man gets mugged in New York. (F ′)
An utterance of A is ambiguous.F and F ′ are unambiguous alternatives for the two possiblereadings of A.Assume that the speaker says A. How to interpret it?
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 61 / 70
Parikh’s Example: Resolving Ambiguities
Meanings
The hearer has to interpret A. Possible meanings are:1. ϕ: Meaning of ‘every ten minutes some man or other gets mugged
in New York.’2. ϕ′: Meaning of ‘Every ten minutes a particular man gets mugged
in New York.’
Possible speaker types are:1. θS = v : State where the speaker knows that ϕ.2. θ′S = w : State where the speaker knows that ϕ′.
Probabilities:1. ρ: Probability of v .2. ρ′ = 1− ρ: Probability of w .
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 62 / 70
Parikh’s Example: Resolving Ambiguities
The Game Tree
u uu
u
u
u
-QQQQQQQs -
PPPPPPPq
1
uu
PPPPPPPq
1
uu
u u--
3
1
1
e
t
e′
t ′
ρ
ρ′
F
A
AF ′
+7
+10
−10
−10
+10
+7
ϕ
ϕ
ϕ′
ϕ
ϕ′
ϕ′
u
*
HHHHHHHj
0v
w
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 63 / 70
Parikh’s Example: Resolving Ambiguities
The Strategies
v wS A AS′ A F ′
S′′ F AS′′′ F F ′
A F F ′
H ϕ ϕ ϕ′
H ′ ϕ′ ϕ ϕ′
The StrategiesSpeaker: S,S′,S′′,S′′′
Hearer: H,H ′
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 64 / 70
Parikh’s Example: Resolving Ambiguities
The Payoffs
v H H ′
S 10 −10S′ 10 −10S′′ 7 7S′′′ 7 7
w H H ′
S −10 10S′ 7 7S′′ −10 10S′′′ 7 7
The PayoffsLeft: In situation v
Right: In situation w
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 65 / 70
Parikh’s Example: Resolving Ambiguities
Expected UtilitiesCase of pure strategies
1. Assumption: Rational players maximise their expected utilities.2. Depends on the probability P(v), the strategies S,H and payoffs.
3. Speaker:
ES(S|H) =∑
v
P(v) uS(v ,S(v),H(S(v))). (6.2)
4. Hearer:EH(H|S) =
∑v
P(v) uH(v ,S(v),H(S(v))). (6.3)
5. In our example: uS = uH , and therefore ES(S|H) = EH(H|S).
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 66 / 70
Parikh’s Example: Resolving Ambiguities
Expected Utilities
H H ′
S 8 −8S′ 9.7 −8.3S′′ 5.3 7.3S′′′ 7 7
The Expected PayoffsProbability of v : ρ = 0.9Probability of w : ρ′ = 0.1
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 67 / 70
Parikh’s Example: Resolving Ambiguities
Analysis
There are two Nash equilibria: (S′,H) and (S′′,H ′).
H H ′
S 8 −8S′ 9.7 −8.3S′′ 5.3 7.3S′′′ 7 7
The first one is also a Pareto Nash equilibrium.
H H ′
S 8 −8S′ 9.7 −8.3S′′ 5.3 7.3S′′′ 7 7
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 68 / 70
Parikh’s Example: Resolving Ambiguities
Assuming that rational players agree on the Pareto Nashequilibrium⇒ they will choose (S′,H).With (S′,H) the utterance A should be interpreted as meaning ϕ:
A: Every ten minutes a man gets mugged in New York.ϕ: Every ten minutes some man or other gets mugged in New York.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 69 / 70
Parikh’s Example: Resolving Ambiguities
The Pareto Nash Solution
u uu
u
u
u
-QQQQQQQs -
PPPPPPPq
1
uu
PPPPPPPq
1
uu
u u--
3
1
1
e
t
e′
t ′
ρ
ρ′
F
A
AF ′
+7
+10
−10
−10
+10
+7
ϕ
ϕ
ϕ′
ϕ
ϕ′
ϕ′
u
*
HHHHHHHj
0v
w
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 70 / 70
Parikh’s Example: Resolving Ambiguities
Literature:
Benz, A., Jäger, G., and van Rooij, R. (2006).An Introduction to Game Theory for Linguists.In Benz, A., Jäger, G., and van Rooij, R., editors, Game Theoryand Pragmatics, pages 1–82. Palgrave Macmillan, Basingstoke.
Binmore, K. (1990).Essays on the Foundations of Game Theory.Basil Blackwell, Cambridge, MA.
Grice, H. P. (1989).Studies in the Way of Words.Harvard University Press, Cambridge MA.
Parikh, P. (2001).The Use of Language.CSLI Publications, Stanford.
Anton Benz (ZAS) GT Pragmatics ESSLLI, 16 August 10 70 / 70